Peculiar Velocities and Kinematic Probes

Peculiar velocity is the motion left after a cosmological distance–redshift relation has been specified. It is therefore not read directly from a redshift: one must combine redshift with an independently calibrated distance indicator, propagate the strongly non-Gaussian distance transformation, and model the survey selection. The reward is unusual sensitivity to the low-redshift velocity field, bulk motion, and the growth of structure. The cost is that calibration, geometry, grouping, and estimator response can imitate the coherent signal one seeks.

Required background. Geodesics and Jacobi Fields supplies the observer-frame projection of a source four-velocity onto a photon direction.

Helpful background. Einstein Equations supplies the field equations underlying the linear velocity–density relation; Relativistic Matter and Compact Objects supplies the astrophysics of standardizable sources; Relativistic Cosmology supplies background distances and linear growth; and Probability and Statistical Inference supplies hierarchical likelihoods, covariance modeling, and posterior checks.

Redshift is not a peculiar-velocity meter

For Doppler motion relative to an FLRW background, with the observer-frame correction already applied and gravitational redshift terms neglected, the exact redshift composition is multiplicative,

$$1+z_{\rm obs}=(1+z_{\rm cos})(1+z_{\rm pec}),$$

not $z_{\rm obs}=z_{\rm cos}+z_{\rm pec}$ beyond first order. Here $z_{\rm pec}$ is the full Doppler factor minus one, including transverse time dilation when relevant. General perturbed spacetimes also contain the endpoint and integrated terms derived on Distance–Redshift Relations with Perturbations. When $\lvert\boldsymbol v\rvert\ll c$, $z_{\rm pec}\simeq v_r/c$ and

$$v_r\simeq c\,\frac{z_{\rm obs}-z_{\rm cos}}{1+z_{\rm cos}}.$$

At very low redshift this reduces to the familiar $v_r\simeq cz-H_0d$. The sign convention here is $v_r\gt0$ for motion away from the observer. Both $z_{\rm obs}$ and the inferred velocity must be assigned to a stated frame—normally heliocentric measurements transformed to the cosmic-microwave-background frame—because an omitted frame conversion produces a sky dipole.

A distance indicator supplies a distance modulus $\mu=5\log_{10}(d_L/10\,{\rm pc})$, or equivalently a log-distance ratio such as

$$\eta\equiv\log_{10}\!\left(\frac{D_z}{D_H}\right),$$

where $D_z$ is the distance assigned by redshift in the chosen background and $D_H$ is the indicator distance. Gaussian scatter in $\mu$ or $\eta$ becomes skewed in linear distance and velocity. Fitting a Gaussian to a naively transformed $v_r$ therefore creates distance-dependent bias. A defensible analysis either remains in log-distance space or forward-models the nonlinear transformation, as emphasized in Watkins et al. 2023, §2.

From radial data to a coherent flow

For a subhorizon, pressureless growing mode with scale-independent growth $D(a)$, the Fourier convention $e^{i\boldsymbol k\cdot\boldsymbol x}$ and continuity equation $\dot\delta+a^{-1}\boldsymbol\nabla\cdot\boldsymbol v=0$ give the irrotational velocity

$$\boldsymbol v(\boldsymbol k,a)=i\,aHf\,\frac{\boldsymbol k}{k^2}\,\delta(\boldsymbol k,a), \qquad f\equiv\frac{d\ln D}{d\ln a}.$$

Only $u_i=\hat{\boldsymbol n}_i\!\cdot\!\boldsymbol v_i$ is observed. For a constant bulk vector $\boldsymbol B$, write $u_i=R_{i\alpha}B_\alpha+\epsilon_i$ with $R_{i\alpha}=\hat n_{i\alpha}$. Generalized least squares gives

$$\widehat{\boldsymbol B} = \left(R^{\mathsf T}C^{-1}R\right)^{-1} R^{\mathsf T}C^{-1}\boldsymbol u.$$

This compact formula hides the central difficulty: $C$ must include distance-indicator errors, nonlinear small-scale dispersion, shared zero points, grouping, and cosmic covariance. In a growth analysis, a representative signal term is

$$\begin{aligned} C^{vv}_{ij}={}&(aHfD)_i(aHfD)_j \int\frac{d^3k}{(2\pi)^3}\, \frac{P_{m,0}(k)}{k^2} (\hat{\boldsymbol k}\!\cdot\!\hat{\boldsymbol n}_i) (\hat{\boldsymbol k}\!\cdot\!\hat{\boldsymbol n}_j) e^{i\boldsymbol k\cdot(\boldsymbol x_i-\boldsymbol x_j)}\\ &+\delta_{ij}\sigma_*^2, \end{aligned}$$

where $D(1)=1$ and $P_{m,0}$ is the present linear matter spectrum. The two growth factors propagate the reference density field to the two source times; one cannot use a single equal-time spectrum for widely separated epochs without them. This covariance is augmented by measurement and calibration terms. A mask-weighted bulk flow, a Wiener-filter reconstruction, and a velocity correlation function are different filtered summaries of the field. Equal quoted radii do not guarantee equal window functions or effective depths.

CF4 data and observational assumptions

Evidence through 2026-08-19 uses the versioned 2023 CF4 paper and dated archival records. Tully et al. report 55,877 galaxies assembled into 38,065 physical groups, while their Tables 3–4 describe 38,057 group entries Tully et al. 2023, abstract and §§2–8; the independent VizieR J/ApJ/944/94 record reports 38,053 rows.

The mutable Extragalactic Distance Database service was inspected on 2026-08-21. At that later inspection its selectors reported 55,874 individual and 38,057 group entries, and its warning rejected the preliminary August 2022 tables. Those live observations document the later access path only and are excluded from the cutoff evidence matrix; no EDD table or page snapshot was frozen. The distinct paper, archive, and live-service counts must not be silently merged: a computation must name and checksum the exact downloaded table.

Calibration. CF4 combines eight methods, dominated by Tully–Fisher and Fundamental Plane distances, with Type Ia supernovae providing overlap and Cepheids, the tip of the red-giant branch, and the NGC 4258 maser anchoring the absolute scale. Bayesian cross-calibration and grouping reduce random scatter, but a common zero-point shift remains coherent; Tully et al. quote a potential systematic of roughly $3\ {\rm km\,s^{-1}\,Mpc^{-1}}$ on their associated $H_0$ estimate. Every likelihood must state the adopted zero point, frame transformation, grouping version, Malmquist correction, and velocity-reconstruction normalization.

Selection. Required cuts include the redshift or distance range, sky mask, indicator-specific signal-to-noise and morphology criteria, group membership, calibrator membership, extinction treatment, and outlier policy. The deeper SDSS contribution is concentrated in the north, whereas the southern sample is shallower; this anisotropy couples survey geometry to a vector-flow estimator. The corrected EDD CF4 table and column description, displaying update date 2023-05-02 and inspected 2026-08-21 exposes the individual method moduli and their errors, which permits method-split checks rather than treating the compilation as homogeneous. This post-cutoff schema observation is a reproduction aid, not evidence about the service state at the scientific cutoff.

Statistical model. A transparent forward likelihood uses the observed $\eta_i$ or $\mu_i$, a latent true distance, the cosmological velocity covariance, calibration offsets with stated priors, an indicator-scatter model, and a nonlinear dispersion $\sigma_*$. The catalog selection must enter the generative model or be validated with selected mocks. Report statistical uncertainty separately from zero-point, grouping, mask, estimator, nonlinear-flow, and cosmic-variance contributions; adding them all to a diagonal error bar loses the correlations that matter most for a bulk vector.

Results and limitations through 2026-08-19

The same CF4 observations support materially different claims because analyses use different bias corrections, windows, mocks, and uncertainty definitions. Whether the largest-scale flow is anomalous remains unresolved. The matrix records results conditional on each stated pipeline, not an unconditional verdict about $\Lambda$CDM.

Evidence or result	Method and regime	Scope or limitation
CF4 corrected catalog: 55,877 galaxy distances from eight methods	Bayesian cross-calibration, grouping, several absolute anchors; $z\lt0.1$	Heterogeneous indicators, anisotropic depth, common zero point; a catalog is not itself a bulk-flow detection
Watkins et al. 2023: increasing large-scale-flow tension	Minimum-variance estimator on CF4 groups, radii $150$–$200\,h^{-1}{\rm Mpc}$	Significance depends on bias correction, ideal-survey window, and quoted uncertainty Watkins et al. 2023, §§4–8
Whitford, Howlett, and Davis 2023: $408\pm165\ {\rm km\,s^{-1}}$ at effective depth $49\,h^{-1}{\rm Mpc}$ with Kaiser MLE	512 realistic CF4 mocks used to diagnose estimator bias and rescale errors	Mocks omit some low-redshift calibrator populations and are model-dependent; inferred flow is not significantly discrepant in that treatment Whitford et al. 2023, §§4.3–6
Hoffman et al. 2024: about $2.6\sigma$ at the matched $150$ and $200\,h^{-1}{\rm Mpc}$ radii	Bias-Gaussianized Wiener-filter/constrained-realization reconstruction using the 2023-05-17 CF4 version	Same data but different bias correction, prior, and smoothing; not directly interchangeable with minimum-variance estimates Hoffman et al. 2024, §6 and Table 1
Wang et al. 2026: $f\sigma_8=0.384^{+0.116}_{-0.194}$ with total uncertainty	Parallel velocity-correlation function, anisotropic mocks, MCMC	Larger and non-Gaussian cosmic variance; reduced $\chi^2\simeq2.2$ and underlying analysis products are available only on request Wang et al. 2026, §§4–6 and Table 2

Direct distance indicators, supernova velocity correlations, kinetic-Sunyaev–Zel’dovich estimators, and density-field reconstructions have distinct calibration and window functions. Agreement across them is stronger evidence than agreement among reweightings of one catalog. Conversely, an “anomalous bulk flow” claim must survive frame changes, depth scans, indicator splits, zero-point shifts, and estimator injection tests. A spatial average of expansion on a hypersurface does not directly equal any of these light-cone measurements; only a forward map through photons, source selection, and the observer frame can establish a limiting correspondence.

What could change the conclusion. The comparison could change when a new homogeneous velocity catalog, a shared recalibration, an end-to-end selection simulation, or a statistically controlled cross-probe inconsistency changes the catalog likelihood or survives matched-window mock tests.

Reproduction path and present boundary

A minimum reproducible workflow is to download one named corrected EDD table, record its retrieval date, checksum, row count, schema, and flags, transform all redshifts into one documented frame, and reproduce a published radial statistic with the authors’ cuts. Then inject known constant vectors through mocks carrying the actual angular and radial selection, recover them with the chosen estimator, and repeat after varying zero points, group assignments, indicator subsets, $\sigma_*$, depth, and covariance. Posterior predictive checks should include sky residual maps, north–south splits, method-split histograms in $\eta$, and the recovered-flow distribution in mocks.

No CF4 catalog snapshot, analysis covariance, or end-to-end mock suite is stored with this page. The EDD schema and primary papers were inspected, but no headline bulk flow or growth constraint was numerically reproduced here. In particular, Wang et al. state that their underlying analysis data are available only on reasonable request. The numerical entries above therefore remain attributed results, not independent replications.

Diagnostic: a small zero-point shift

Suppose an indicator assigns $d=100\,{\mathrm{Mpc}}$ and a calibration revision shifts every distance modulus by $\Delta\mu=+0.05$ mag. At fixed CMB-frame redshift and $H_0=70\ {\rm km\,s^{-1}\,Mpc^{-1}}$, compute the change in the inferred low-redshift radial peculiar velocity. Should the shift enter as independent noise or as a shared nuisance parameter?

Solution

The distance ratio is

$$\frac{d'}{d}=10^{\Delta\mu/5},$$

so the new distance is $d'=100\times10^{0.01}=102.33\,{\mathrm{Mpc}}$. Holding the redshift fixed, the low-redshift peculiar velocity changes by

$$\Delta v_r=-H_0(d'-d)=-163\ {\rm km\,s^{-1}}.$$

That displacement is comparable to the signal assigned to large-scale motion, and it is coherent for every object sharing the calibration. In an isotropic sample it mainly produces a radial monopole; an anisotropic survey can mix that pattern into the fitted bulk vector. Treating $0.05$ mag as independent scatter loses this shared structure. The diagnostic is to inject the offset as one shared nuisance parameter and verify the response of the inferred flow under the actual survey window.

References

• Extragalactic Distance Database. 2023. “All CF4 Individual Distances: Column Descriptions.” University of Hawaiʻi Institute for Astronomy, updated 2023-05-02. Catalog schema.
• Hoffman, Yehuda, Aurélien Valade, Noam I. Libeskind, Jenny G. Sorce, R. Brent Tully, Simon Pfeifer, Stefan Gottlöber, and Daniel Pomarède. 2024. “The Large-Scale Velocity Field from the Cosmicflows-4 Data.” Monthly Notices of the Royal Astronomical Society 527: 3788–3805. DOI.
• Tully, R. Brent, et al. 2023. “Cosmicflows-4.” The Astrophysical Journal 944: 94. DOI. Corrected public catalogs.
• Wang, Yuyu, Hume A. Feldman, Richard Watkins, and Xiaohu Yang. 2026. “The Peculiar Velocity Correlation Function of the Cosmicflows-4 Catalogue.” Monthly Notices of the Royal Astronomical Society 548: stag677. DOI.
• Watkins, Richard, et al. 2023. “Analysing the Large-Scale Bulk Flow Using Cosmicflows4: Increasing Tension with the Standard Cosmological Model.” Monthly Notices of the Royal Astronomical Society 524: 1885–1892. DOI.
• Whitford, Abbé M., Cullan Howlett, and Tamara M. Davis. 2023. “Evaluating Bulk Flow Estimators for CosmicFlows–4 Measurements.” Monthly Notices of the Royal Astronomical Society 526: 3051–3071. DOI.